3.5.71 \(\int \frac {\arcsin (\sqrt {1+b x^2})^n}{\sqrt {1+b x^2}} \, dx\) [471]

3.5.71.1 Optimal result

Integrand size = 26, antiderivative size = 38 \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^2} \arcsin \left (\sqrt {1+b x^2}\right )^{1+n}}{b (1+n) x} \]

output

arcsin((b*x^2+1)^(1/2))^(1+n)*(-b*x^2)^(1/2)/b/(1+n)/x
 

3.5.71.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^2} \arcsin \left (\sqrt {1+b x^2}\right )^{1+n}}{b (1+n) x} \]

input

Integrate[ArcSin[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]
 

output

(Sqrt[-(b*x^2)]*ArcSin[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)
 

3.5.71.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5333, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arcsin \left (\sqrt {b x^2+1}\right )^n}{\sqrt {b x^2+1}} \, dx\)

\(\Big \downarrow \) 5333

\(\displaystyle \frac {\sqrt {-b x^2} \int \frac {\arcsin \left (\sqrt {b x^2+1}\right )^n}{\sqrt {-b x^2}}d\sqrt {b x^2+1}}{b x}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\sqrt {-b x^2} \arcsin \left (\sqrt {b x^2+1}\right )^{n+1}}{b (n+1) x}\)

input

Int[ArcSin[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]
 

output

(Sqrt[-(b*x^2)]*ArcSin[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)
 

3.5.71.3.1 Defintions of rubi rules used

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5333

Int[ArcSin[Sqrt[1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[1 + (b_.)*(x_)^2], x_Symbol] 
:> Simp[Sqrt[(-b)*x^2]/(b*x)   Subst[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, 
Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]
 

3.5.71.4 Maple [F]

\[\int \frac {\arcsin \left (\sqrt {b \,x^{2}+1}\right )^{n}}{\sqrt {b \,x^{2}+1}}d x\]

input

int(arcsin((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x)
 

output

int(arcsin((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x)
 

3.5.71.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\frac {\sqrt {-b x^{2}} \arcsin \left (\sqrt {b x^{2} + 1}\right )^{n} \arcsin \left (\sqrt {b x^{2} + 1}\right )}{{\left (b n + b\right )} x} \]

input

integrate(arcsin((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="fricas")
 

output

sqrt(-b*x^2)*arcsin(sqrt(b*x^2 + 1))^n*arcsin(sqrt(b*x^2 + 1))/((b*n + b)* 
x)
 

3.5.71.6 Sympy [F]

\[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\begin {cases} \frac {2 x}{\pi } & \text {for}\: b = 0 \wedge n = -1 \\x \left (\frac {\pi }{2}\right )^{n} & \text {for}\: b = 0 \\\int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {asin}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx & \text {for}\: n = -1 \\\frac {\sqrt {- b x^{2}} \operatorname {asin}{\left (\sqrt {b x^{2} + 1} \right )} \operatorname {asin}^{n}{\left (\sqrt {b x^{2} + 1} \right )}}{b n x + b x} & \text {otherwise} \end {cases} \]

input

integrate(asin((b*x**2+1)**(1/2))**n/(b*x**2+1)**(1/2),x)
 

output

Piecewise((2*x/pi, Eq(b, 0) & Eq(n, -1)), (x*(pi/2)**n, Eq(b, 0)), (Integr 
al(1/(sqrt(b*x**2 + 1)*asin(sqrt(b*x**2 + 1))), x), Eq(n, -1)), (sqrt(-b*x 
**2)*asin(sqrt(b*x**2 + 1))*asin(sqrt(b*x**2 + 1))**n/(b*n*x + b*x), True) 
)
 

3.5.71.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(arcsin((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="maxima")
 

output

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imagi 
nary; found sqrt(-_SAGE_VAR_b)
 

3.5.71.8 Giac [F]

\[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\int { \frac {\arcsin \left (\sqrt {b x^{2} + 1}\right )^{n}}{\sqrt {b x^{2} + 1}} \,d x } \]

input

integrate(arcsin((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="giac")
 

output

sage0*x
 

3.5.71.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin \left (\sqrt {1+b x^2}\right )^n}{\sqrt {1+b x^2}} \, dx=\int \frac {{\mathrm {asin}\left (\sqrt {b\,x^2+1}\right )}^n}{\sqrt {b\,x^2+1}} \,d x \]

input

int(asin((b*x^2 + 1)^(1/2))^n/(b*x^2 + 1)^(1/2),x)
 

output

int(asin((b*x^2 + 1)^(1/2))^n/(b*x^2 + 1)^(1/2), x)